Let $D$ be the midpoint of the arc $B-A-C$ of the circumcircle of $\triangle ABC (AB<AC)$. Let $E$ be the foot of perpendicular from $D$ to $AC$. Prove that $|CE|=\frac{|BA|+|AC|}{2}$.

Prove for all positive real numbers $a,b,c$, such that $a^2+b^2+c^2=1$: \[\frac{a^3}{b^2+c}+\frac{b^3}{c^2+a}+\frac{c^3}{a^2+b}\ge \frac{\sqrt{3}}{1+\sqrt{3}}.\]

Prove that for all odd prime numbers $p$ there exist a natural number $m<p$ and integers $x_1, x_2, x_3$ such that: \[mp=x_1^2+x_2^2+x_3^2.\]

Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.

Given is a triangle $\triangle ABC$ and points $M$ and $K$ on lines $AB$ and $CB$ such that $AM=AC=CK$. Prove that the length of the radius of the circumcircle of triangle $\triangle BKM$ is equal to the lenght $OI$, where $O$ and $I$ are centers of the circumcircle and the incircle of $\triangle ABC$, respectively. Also prove that $OI\perp MK$.

A unit square is divided into polygons, so that all sides of a polygon are parallel to sides of the given square. If the total length of the segments inside the square (without the square) is $2n$ (where $n$ is a positive real number), prove that there exists a polygon whose area is greater than $\frac{1}{(n+1)^2}$.